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trolling geometry problem
iStud 3
N
by iStud
Source: Monthly Contest KTOM April P3 Essay
Given a cyclic quadrilateral 
with 
and 
. Lines 
and 
intersect at 
, and lines 
and 
intersect at 
. Let 
be the midpoints of sides 
, respectively. Let 
and 
be points on segment 
and 
, respectively, so that 
is the angle bisector of 
and 
is the angle bisector of 
. Prove that 
is parallel to 
if and only if 
divides into two triangles with equal area.
with 
and 
. Lines 
and 
intersect at 
, and lines 
and 
intersect at 
. Let 
be the midpoints of sides 
, respectively. Let 
and 
be points on segment 
and 
, respectively, so that 
is the angle bisector of 
and 
is the angle bisector of 
. Prove that 
is parallel to 
if and only if 
divides into two triangles with equal area.
3 replies
3D geometry theorem
KAME06 1
N
by mathuz
Let 
a point in the space and 
the centroid of a tetrahedron . Prove that:
a point in the space and 
the centroid of a tetrahedron . Prove that:
1 reply
IMO Shortlist 2012, Number Theory 6
mathmdmb 42
N
by ihategeo_1969
Source: IMO Shortlist 2012, Number Theory 6
Let 
and 
be positive integers. If 
is divisible by 
for every positive integer 
, prove that 
.
and 
be positive integers. If 
is divisible by 
for every positive integer 
, prove that 
.
42 replies
GCD of a sequence
oVlad 7
N
by grupyorum
Source: Romania EGMO TST 2017 Day 1 P2
Determine all pairs 
of positive integers with the following property: all of the terms of the sequence 
have a greatest common divisor 
of positive integers with the following property: all of the terms of the sequence 
have a greatest common divisor 
7 replies
Another System
worthawholebean 3
N
by P162008
Source: HMMT 2008 Guts Problem 33
Let 
, 
, 
be nonzero real numbers such that 
and 
. Find the value of
.
, 
, 
be nonzero real numbers such that 
and 
. Find the value of
.
3 replies
Inequality with three conditions
oVlad 2
N
by Quantum-Phantom
Source: Romania EGMO TST 2019 Day 1 P3
Let 
be non-negative real numbers such that ![\[b+c\leqslant a+1,\quad c+a\leqslant b+1,\quad a+b\leqslant c+1.\]](//latex.artofproblemsolving.com/b/9/e/b9e86e898c0536b7323a03611d5bdbf679caa710.png)
Prove that 
be non-negative real numbers such that ![\[b+c\leqslant a+1,\quad c+a\leqslant b+1,\quad a+b\leqslant c+1.\]](http://latex.artofproblemsolving.com/b/9/e/b9e86e898c0536b7323a03611d5bdbf679caa710.png)
Prove that 
2 replies
GCD Functional Equation
pinetree1 61
N
by ihategeo_1969
Source: USA TSTST 2019 Problem 7
Let 
be a function satisfying
for all integers and . Show that there exist positive integers 
and such that 
for all integers .
Ankan Bhattacharya
be a function satisfying
for all integers and . Show that there exist positive integers 
and such that 
for all integers .Ankan Bhattacharya
61 replies
An easy FE
oVlad 3
N
by jasperE3
Source: Romania EGMO TST 2017 Day 1 P3
Determine all functions 
such that ![\[f(xy-1)+f(x)f(y)=2xy-1,\]](//latex.artofproblemsolving.com/8/8/8/888ca39f2b7f8cec6d6426bee28d40eade40a66e.png)
for any real numbers 
and 
such that ![\[f(xy-1)+f(x)f(y)=2xy-1,\]](http://latex.artofproblemsolving.com/8/8/8/888ca39f2b7f8cec6d6426bee28d40eade40a66e.png)
for any real numbers 
and 
3 replies
p^3 divides (a + b)^p - a^p - b^p
62861 49
N
by Ilikeminecraft
Source: USA January TST for IMO 2017, Problem 3
Prove that there are infinitely many triples 
of positive integers with 
prime, 
, and 
, such that 
is a multiple of 
.
Noam Elkies
of positive integers with 
prime, 
, and 
, such that 
is a multiple of 
.Noam Elkies
49 replies
Funny easy transcendental geo
qwerty123456asdfgzxcvb 1
N
by golue3120
Let 
be a logarithmic spiral centered at the origin (ie curve satisfying for any point on it, line 
makes a fixed angle with the tangent to at ). Let 
be a rectangular hyperbola centered at the origin, scaled such that it is tangent to the logarithmic spiral at some point.
Prove that for a point
on the spiral, the polar of wrt. is tangent to the spiral.
be a logarithmic spiral centered at the origin (ie curve satisfying for any point on it, line 
makes a fixed angle with the tangent to at ). Let 
be a rectangular hyperbola centered at the origin, scaled such that it is tangent to the logarithmic spiral at some point.Prove that for a point
on the spiral, the polar of wrt. is tangent to the spiral.
1 reply
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